MTH 525, FALL 2006

Brief Notes for Weeks 7 & 8

(very tentative)

Monday, October 9

We begin the study of determinants (chapter 5.)

We define ring and n-linear functions on matrices over a ring. We show that a linear combination of n-linear functions in n-linear (Lemma, p. 143.) We define alternating functions and then determinant functions and some immediate properties of these functions. We then classify determinant functions in Theorem 1 and show that determinant functions exist.

Do p. 148: 1, 3, 4.

Wednesday, October 11

Assignment 5 is collected.

We continue with the study of determinants, introducing permutations (section 5.3) and showing that the determinant is unique.

Do p. 155: 1, 2, 3.

We prove that the determinant is multiplicative (Theorem 3.) We introduce the adjoint and prove additional properties of the determinant (section 5.4.)

We relate the determinant to elementary row operations.

Do p. 162: 1, 2, 3, 4, 5. (Problems 3, 4, 5 deal with skew-symmetric, orthogonal and unitary matrices.)

Monday, October 16

The midterm exam is given. This exam is over chapters 1-4 (and is 200 points.)

Wednesday, October 18

We finish chapter 5. We define modules over a ring (section 5.5), alternating multilinear forms on modules (section 5.6) and introduce the Grassman ring (section 5.7.)

Last modified October 1, 2006